Jeffrey D. Ullman Stanford University

2  Generalizes: 1.Moving loop-invariant computations outside the loop. 2.Eliminating common subexpressions. 3.True partial redundancy: an expression is available along some paths to a point, but not along others.

3  Throughout, assume that neither argument of an expression x+y is modified unless we explicitly assign to x or y.  And of course, we assume x+y is the only expression anyone would ever want to compute.

4 t = x+y a = x+y a = t

5 a = x+yb = x+y c = x+y t = x+y a = t t = x+y b = t c = t

6 a = x+y b = x+y t = x+y a = t t = x+y b = t

7  We could: 1.Add a new block along an edge.  Only necessary if the edge enters a block with several predecessors. 2.Duplicate blocks so an expression x+y is evaluated only along paths where it is needed.

8 t = x+y = t = x+y t = x+y

9  Can exponentiate the number of nodes.  Our PRE algorithm needs to move code to new blocks along edges, but will not split blocks.  Convention: All new instructions are either inserted at the beginning of a block or placed in a new block.

10 1. Determine for each expression the earliest place(s) it can be computed while still being sure that it will be used. 2. Postpone the expressions as long as possible without introducing redundancy.  Trade space for time – an expression can be computed in many places, but never if it is already computed.  Guarantee: No expression is computed where its value might have been computed previously.

3. Determine those places where it is really necessary to store x+y in a temporary rather than compute it when needed.  Example: If x+y is computed in only one place, then we do not want to compute it early and store it. 11

12  We use four data-flow analyses, in succession, plus some set operations on the results of these analyses.  After the first, each analysis uses the results of the previous ones in a role similar to that of Gen (for RD’s) or Use (for LV’s).

13  Expression x+y is anticipated at a point if x+y is certain to be evaluated along any computation path, before any recomputation of x or y.  An example of the fourth kind of DF schema: backwards-intersection.

14 = x+y x+y is anticipated here and could be computed now rather than later. = x+y x+y is anticipated here, but is also available. No computa- tion is needed.

15  Use(B) = set of expressions x+y evaluated in B before any assignment to x or y.  Def(B) = set of expressions one of whose arguments is assigned in B.  Direction = backwards.  Meet = intersection.  Boundary condition: IN[exit] = ∅.  Transfer function: IN[B] = (OUT[B] – Def(B)) ∪ Use(B)

16 = x+y Anticipated Backwards; Intersection; IN[B] = (OUT[B] – Def(B)) ∪ Use(B)

17  Modification of the usual AE.  x+y is “available” at a point if either: 1.It is available in the usual sense; i.e., it has been computed and not killed, or 2.It is anticipated; i.e., it could be available if we chose to precompute it there.

18  x+y is in Kill(B) if x or y is defined, and x+y is not recomputed later in B (same as for earlier definition of “available expressions”).  Direction = Forward  Meet = intersection.  Transfer function: OUT[B] = (IN[B] ∪ IN ANTICIPATED [B]) – Kill(B)

19  x+y is in Earliest[B] if it is anticipated at the beginning of B but not “available” there.  That is: when we compute anticipated expressions, x+y is in IN[B], but  When we compute “available” expressions, x+y is not in IN[B].  I.e., x+y is anticipated at B, but not anticipated at OUT of some predecessor.

20 = x+y Anticipated “Available” Earliest = anticipated but not available Forward; Intersection; OUT[B] = (IN[B] ∪ IN ANTICIPATED [B]) – Kill(B)

21  Now, we need to delay the evaluation of expressions as long as possible, but … 1.Not past the use of the expression. 2.Not so far that we wind up computing an expression that is already evaluated.  Note viewpoint: It is OK to use code space if we save register use.

22  x+y is postponable to a point p if on every path from the entry to p: 1.There is a block B for which x+y is in earliest[B], and 2.After that block, there is no use of x+y.

23  Computed like “available” expressions, with two differences: 1.In place of killing an expression (assigning to one of its arguments): Use(B), the set of expressions used in block B. 2.In place of IN ANTICIPATED [B]: earliest[B].

24  Direction = forward.  Meet = intersection.  Transfer function: OUT[B] = (IN[B] ∪ earliest[B]) – Use(B)

25 = x+y Earliest Postponable Three places to compute x+y Forward; Intersection; OUT[B] = (IN[B] ∪ earliest[B]) – Use(B)

26  We want to postpone as far as possible.  Question: Why?  How do we compute the “winners” – the blocks such that we can postpone no further?  Remember – postponing stops at a use or at a block with another predecessor where x+y is not postponable.

27  For x+y to be in latest[B]: 1.x+y is either in earliest[B] or in IN POSTPONABLE [B].  I.e., we can place the computation at B. 2.x+y is either used in B or there is some successor of B for which (1) does not hold.  I.e., we cannot postpone further along all branches.

28 = x+y Earliest Or Postponable to beginning Used Or has a suc- cessor not red. Latest = gold and red.

29  We’re now ready to introduce a temporary t to hold the value of expression x+y everywhere.  But there is a small glitch: t may be totally unnecessary.  E.g., x+y is computed in exactly one place.

30  used[B] = expressions used along some path from the end of B.  Direction = backward.  Meet = union.  Transfer function: IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B)  e-used = “expression is used in B.”

31 = x+y Recall: Latest Used Backwards; Union; IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B)

32 1. If x+y is in both Latest[B] and OUT USED [B], introduce t = x+y at the beginning of B. 2. If x+y is used in B, but either 1.Is not in Latest[B] or 2.Is in OUT USED [B], replace the use(s) of x+y by uses of t.

33 = x+y Recall: Latest Recall OUT USED Create temp- orary here Use it here But not here --- x+y is in Latest and not in OUT USED

34 = x+y t = x+y = t t = x+y